In Exercises 52–55, use Cramer’s Rule to solve each system.

Question

In Exercises 52–55, use Cramer’s Rule to solve each system. <Image>

Accepted Answer

Welcome back. I'm so glad you're here. We've got a system of equations we're going to solve using Cramer's rule. So what is Cramer's rule? Well, you can recall from previous lessons, the Cramer's rule says that our x value is going to be equal to the determinant of X. Over our system of equations determinant and our Y value is going to equal our determinant of why over the determinant of our system of equations, that's going to give us our x and Y values. It's important to note that the determinant cannot equal zero because that would make it undefined. So what is our determinant? Well, you can recall from previous lessons that are determinant is going to be are a one, going down the column A one and A two And RB one and B two. What does that mean? Well, in our system of equations we can create a matrix and for our X values that will be our A one and a two. R Y values will be our B one and B two and our constance will be our C one and C two. And for our determinant for our x values, we're going to replace the X value coefficients with the constant terms C one and C two. And we keep our Y value coefficients R B one and B two. And very similarly for our wide determinant, we keep our X is a one and A two and we replace our Y values with the constant terms C one and C two. Now let's just start filling things in if we were to create a matrix based on our system of equations, we would have two for the X coefficient six for the y coefficient and two for our constant. One for the second. Rose X coefficient five for the second, Rose y coefficient and negative five for the second. Rose constant. Now we can just start to plug things in so let's sulfur are determinant. Make sure it's not equal to zero. The determinant, okay, we take our a. one Which is two below that are a two, which is one. And the second column is R. B one which is six and B two which is five. Now we take our A one times R B 22 times five is 10 minus. We take our A two times R B one, one time six is six or Yes, one time six is six and 10 minus six is four. So our determinant But the system of equations is four one down. Let's get our determinant of our x values. So again, we're replacing those X values and so instead of A one now we've got C one which is two and C two which is negative five. And bringing back our Y values are B one is six and R B two is five. We multiply RC one times RB 2, 2 times five is 10. We multiply RC two times are B one, we subtract negative five times six is negative 30 So 10 -20 is 40. The determine of our x values determinant is 40. Now let's get our Y values determinant. We keep our X values are a one in our A. Two so that's two in one and our Y values are replaced by our constant terms. So that is C one is two and C two is negative five. We're going to multiply our A one times our C 22 times negative five is negative. 10 subtract are a two times R C 11 times two which is two, So -10 -2 is -12. The determinant for AY Value is -12. We've got all three determinants. Now let's plug those back in. For Cramer's rule. We know our X value is going to be the X determinant. D sub X. Which is 40 Divided by the determinant which is four. So our x value is going to be 10. Now let's look at our Y value Y equals R. D sub Y. Ry determinant which is negative 12 divided by our determinant which is four. So our Y value is going to equal negative three. We look at our answer choices and this matches with answer choice. B Well done. We'll catch you on the next one